# Damped Spherical Pendulum

Damped Spherical Pendulum

This Demonstration traces the path of the bob on a damped spherical pendulum. The pendulum is suspended at the center of an imaginary sphere that marks the outer bounds of the center of the bob.

The equations of motion are

gmsin(θ(t))+Lm(t)-Lmsin(θ(t))cos(θ(t))(t)+Lμ(t)0

′′

θ

2

′

ϕ

′

θ

2Lm(t)sin(θ(t))cos(θ(t))(t)+Lm(θ(t))(t)+Lμ(θ(t))(t)0

′

θ

′

ϕ

2

sin

′′

ϕ

2

sin

′

ϕ

where and are the spherical coordinates of the center of gravity of the bob. The pendulum rod has length (with no loss of generality) and bob mass . The damping coefficient of the system is . The initial angular positions are and and the initial angular speeds are and .

θ

ϕ

L=1

m

μ

θ

0

ϕ

0

′

θ

0

′

ϕ

0